However, as a disadvantage, the lengths of bars and joints of shells being subdivided fairly evenly into triangular faces, including Geodesic cupolas (U.S. Pat. No. 2,682,235), all together are large in relation to the envelope's surface. Besides, a lot of cutting waste arises during the production of the small triangular components. That is why shells of meshes having more than three corners are more favorable. Besides, the surface especially of these shells, having the same number of component faces, seems to be less rough, because the vertices, where several facets meet as plane meshes, are more obtuse.
Geodesic domes consist of usually triangular meshes within triangular shell's pieces with sides on net-lines in planes of great circles crossing each other. Usually, the side planes of their pieces include the edges of an imagined icosahedron. If, instead of this, they include the edges of a stellated cube as a determining polyhedron, the typical fine subdivision within a triangles grid of circle arc segments is extraordinarily irregular because of the strong curvature of the shell's pieces being extraordinarily non-equilateral triangular here. (P. Huybers, G. van der Ende: Polyhedral Sphere Subdivisions; in: G. C. Giuliani (Ed.): Spatial Structures: Heritage, Present and Future, International Association for Shell and Spatial Structures International Symposium, Milano, 1995, pp. 189, FIG. 13).
Triangular caps being formed of an eighth of a sphere being subdivided in such a manner, and having right-angled corners are implemented into computer programs in order to round off ashlars. Hitherto, meshes having a quadrangular form are unknown as an alternative for this. During modeling of computer programs by means of splines or nurbs or by two arrays of curves crossing at right angles to each other within the surface, the rounding-off of ashlars including cambering the remained plane faces results into not-plane meshes and causes a large amount of data.
Because of the acuteness of the meshes, a geodesic icosahedral dome of small plane quad rhombus or kite shaped meshes, each consisting of two combined adjoining coplanar component faces of two neighbored approximately equilateral triangular meshes having been stellated to result into very flat pyramids, has the same disadvantages like those consisting of the initial triangular meshes.
Beside the geodesically subdivided spheres, there are convex sphere-like polyhedra, consisting of a plurality of faces, of which faceted shells can be formed, which, by their manifold symmetries, recall geodesic domes but consist of more compact quad faces: so-called “Duals of Transpolyhedra” (H. Lalvani: Transpolyhedra|Dual Transformations by Explosion—Implosion, Papers in Theoretical Morphology 1, Published by Heresh Lalvani, New York, 1977, Library of Congress Card Number: 77-81420, FIG. 8 of “plate 27” on p. 67, p. 19 and p. 60).
However, the plane quad-meshes of the “Dual” of FIG. 8 in “Plate 27” on p. 67 never would be able to be curved in a definite manner keeping the node points unchanged, because this faceted shell has been developed by a recurring, all over happening insertion of new component faces and by a subsequent non-recurring removing all over of old, likewise always flat component faces. Besides, because of this developing process being rendered as an example ibidem on p. 19 and p. 60, the degree of subdivision of a faceted shell's piece, which is as small as possible, is situated within a spatial sector region of the polyhedron, and is repeated by mirroring, turning, and copying, is not free but restricted to 2, 4, 8, 16, . . . . Last but not least, each mesh-quad of such a unit has only disparallel sides (hardly to be recognized at some local areas of the hand-made drawings of Lalvani). As a consequence, there are: a lot of varying edge-lengths, a fundamentally irregular appearance, and a few options for reshaping. That is why not any built Duals of Transpolyhedra are known.
For the construction of assembled shells, translational shells, which, in a faceted design, consist of a plurality of parallelograms, are less wide-spread as geodesic domes. In contrast to the objects described before, they enable synclastic and anticlastic areas equally.
As a disadvantage, they are comparatively flat. Cause, within areas being more steeply sloped, the mesh-quads would be very acute-angled and stretched even within a very symmetrical, regular, that is, circularly round shape in plan of the shell as rotational paraboloid, whereby the portion of joints would be large likewise, and acute special nodes would have to be designed.
Yet unfortunately, even in the case of a flat rotation paraboloid, the advantage to have edge-lengths being as equal as possible is diminished by special lengths on the plane bearing border of the shell as well as by irregular cut-offs of faces there, which make arise accidentally and arbitrarily triangles and pentagons again.
Compared with conventional translational surfaces, so-called “scale-trans surfaces” enlarge the options to shape double-curved shells. (Annette Bögle: “weit breit-Netzschalen/floating roofs—Grid Shells” in: A. Bögle, P. Cachola Schmal, I. Flagge (Ed.): “leicht weit—Light Structures—Jörg Schlaich, Rudolf Bergermann”, (exhibition of the DAM Frankfurt, 2004), Munich, 2003, p. 113-129; Hans Schober: “Glasdächer und Glasfassaden/Glass Roofs and Glass Facades” in: Sophia and Stefan Behling (Ed.): “Glas—Konstruktion und Technologie in der Architektur/Glass—Structure and Technology in Architecture”, Munich, 1999, p. 68-73).
In contrast to a translational surface, shells of one scale-trans surface also enable in some cases shells that join to the ground in a steeply sloped or an outwards inclined way and that are at least in parts synclastic, hereinafter called “blobs”, such as the upper shell 1 in FIG. 1 according to the state of the art, joining perpendicularly to the ground, and having a perpendicular plane opening 2, which is rigid in a self-supporting manner by an anticlastically curved free border region of the shell, hereinafter called “enlargement”. The shape of the shell recalls the surface of a drop of water or oil hanging on a tap and starting to drip off, although being halved, turned 90°, and distorted.
In FIG. 1, the problematic nature of shells with a scale-trans subdivision into meshes is rendered comprehensively in order to distinguish a corresponding shell like in FIG. 151 according to the present invention in the best mode as clear and comparable as possible from the state of the art.
In contrast to translational shells, the upper shell 1 in FIG. 1 doesn't need to have meshes being arbitrarily cut off. A quadrilateral section 3 of such a shell, hereinafter called “ordinary quad section”, will be described more in detail: It is located between four corner points 4, 5, 6, 7. It has a geometric net of two crossing arrays of long curved lines. These curves are aligned with the courses of two side's curves 8, 9, hereinafter called “array sides,” that connect the corner points. Said long curved lines 10, 11, hereinafter called “array curves”, form, in any desired quantity, quadrangular areal meshes 12, hereinafter called, “quad-meshes”, whose corners are fourfold node points 13 being connected by straight lines 14, 15, hereinafter called “chords”. Each time, four chords form the sides of a mesh. Only once, two opposites 14 of them are parallel. Thus, a plane mesh has the shape of a trapezoid. Mostly, the array curves are implemented in a structure as polygonal lines between a plurality of flat meshes having infills of a plane material, whereby the chords are sections of the polygonal line. As a shrinking in this context, the centric “stretching” (German: strecken=scale up) of the array side 8 that is located within a vertical plane resulted into array curves 10 of different sizes but equal shapes each time. It was proceeded from up reference-points within the spatially curved central reference-line 16.
Within the drawings, all curved sides 14, 15 of a mesh are replaced by chords 14, 15. As an exception, these side-curves are rendered only once also in a curved way in the case of an magnified mesh on the left
It is possible to replace the chords by curved lines again, unless a faceted surface is meant explicit and exclusively. The flat curves between the node points 13 as endpoints of parallel chords 14 having differing lengths have the equal shape in a differing size.
The array curves of one orientation having parallel chords 14 are plane. Planes that include a plane array curve are called “array-curve planes” hereinafter. Here, these planes are parallel to the vertical plane of the array side 8. They slice here the shell like an egg-cutter the surface of an peeled egg. In the ground plane, they are represented by arrays of parallel straight lines like the cutter's cutting wires.
As a disadvantage of scale-trans shells, the array curves 11 running side by side but not in parallel, in one of both expansion directions of the net being generated here by a ruled scaling in centrepoints of the central reference-line 16, called “centric” by J. Schlaich and H. Schober, concur on their ends in one sole point 17, like the meridians of a globe on the pole or the ribs of a cupola on the zenith do. This causes yet regular but extremely acute angled triangular meshes 18 being hereby unfavorable here again. It was possible to mitigate this disadvantage only by drawing down the “pole” out of the shell surface and beyond the border at the ground plane. This has happened already to the shell 1. This causes once again cut meshes 19 having a triangular, quadrangular or pentagonal shape. The cut-off region below the x-y-ground plane is rendered by dashed lines.
Consequently, hitherto, the known realized shells of one scale-trans surface are restricted to mainly convex, mostly oblong, mostly synclastic examples being developed along a central reference-line 16. In contrast to the rather spherical shell 1 represented here, these shells have an opening 2 not only at one side, but they are open at two opposite shorter sides.
Hitherto, In the field of building construction, four kinds of shell-forms are able to be produced even of one scale-trans surface only insufficiently or not at all: Bohemian domes, “cushion-roofs”, “inverted suspension-shells”, or so-called “blobs”. In respect of each of these four kinds of shell-forms, this fact will be showed section by section:
Bohemian domes are cupolas having border-arcs each being in a vertical plane. They are possible only above a quadrat, rectangle, parallelogram, or trapezoid, but not above any given straight-lined polygonal outline in plan. In the case of a four-sided plan, two sides have to be parallel. A Bohemian dome above an asymmetrical trapezoid must be generated by a scale-trans subdivision whereat each mesh or, at least in the case of alternation by meshes having the shape of a parallelogram, each second mesh occurs only once as a format. Another plans that are not quadrangular can arise only by dividing diagonally some meshes assigned to the border. If there are more than four corners, two sides have to be parallel once again. Even rectangular Bohemian domes have much formats of meshes.
“Cushion-roofs” are mainly synclastic shells being linearly supported upon a polygon and having anticlastically curved corner areas. (on formfinding of cushion-roofs, see: K. Bach, B. Burkhardt, F. Otto: Mitteilungen des Instituts für leichte Flächentragwerke, No. 18, (IL 18) Seifenblasen/Forming Bubbles, Stuttgart, 1987, p. 234, 235, FIGS. 22 and 25): Cushion-roofs of not twisted quad-meshes having coplanar vertices were not be assumed to be possible at all (H. Schober, p. 69, 70). The built roofs are cambered slightly only in order to avoid strongly twisted meshes. That is why they have to be trussed by cables below. Hitherto, cushion-roofs of twisted quad-meshes always have only four sides, as a rectangle or a square. Higher cushion-roofs are triangulated and show strong folds within the small corner areas probably being anticlastically curved.
“Inverted suspension-shells” are mainly synclastic shells with corner supports and only slight enlargements on the regions of the border-arcs being rigidified and self-supporting by their anticlastic curvature, like several shells of Heinz Isler (E. Heinle, J. Schlaich: Kuppeln, Stuttgart, 1996, p. 187, Fig. below, l. a. r., p. 222, picture. 94): These shells are not feasible in a pure scale-trans subdivision, because the inverse curvature in the cross-section of an anticlastically bent-up boundary region of the predominantly synclastic shell would cause an inverse curvature in the curve of the border-arc of a bent-up boundary region being situated in the transverse direction. This applies also to the next kind of shell-forms:
“Blobs” may have open enlargements in order to form floating transitions (C1-transitions) not only into convex shells being situated one behind the other but into ones being located in various directions, in a manner being comparable to the surface of a drop of water or oil having been drawn apart upon a horizontal repellent plane surface by an acute tool into any diverse directions and hereby being connected to several other drops (for example: Multihalle Mannheim (E. Heinle, J. Schlaich: Kuppeln, Stuttgart, 1996, S. 169). Yet, the irregularly shaped Multihalle has twisted meshes. Hitherto, it would not be able to be produced approximately of elements between coplanar node points.
The problems dealing with blobs are discussed more thoroughly within the next sections:
Hitherto, the number of enlargement openings of blobs having a scale-trans subdivision is four at the most, if acute angled meshes are taken into account. One pair of opposite straight lines of openings or of plane borders in the plan has to be parallel here once again. In the case of a translational shell this concerns two pairs. It is not possible to achieve that straight lines, being aligned with these lines in plan, form an equilateral triangle or an equilateral or irregular polygon having more than four sides in plan. Even a pure translational cupola having four enlargements in a symmetric shape already would have the problem that high openings would implicate very large zenith's heights.
One sole enlargement is easy to implement indeed, as the scale-trans shell 1 shows in FIG. 1. If another enlargement having another plane opening 2 being situated not on the rear but on the left side and joining at a corner is needed, the shell must be changed into a shell 20 shown in FIG. 1 (bottom), having been modified within its left half. In principle, its additional enlargement can be produced only if the orientation of the array-curve's parallel planes being represented in a projection into the ground-plane, which is shown as lying below it, by a plurality of parallel straight lines each crossing several node points is turned 90° within the left half of the shell. Hereby, the shell is no longer homogeneous, but consists of two shell's regions: one on the left, the other on the right of the dividing curve 21 that was able to become plane by extending existing meshes in order to figure a new array side having been movable in parallel into the left direction and having been scalable.
But the border-arc 22 of the new additional opening 2 on the shell's left side is not useful because it joins too flatly to the ground. Besides, the old opening consequently having been changed has a statically unfavorable section 23 of the border-arc. This section has a destabilizing inverse curvature being caused directly by the bending-up of the surface, which initially has been synclastic within this region, to result into an enlargement. Intrinsically, said bending-up had been intended to stabilize the new opening.
Although the scaling in the left half of the shell is not regular, but an irregular, intuitive, and time-consuming stretching mesh by mesh, the degree of ductility has been too low to avoid this inverse curvature within the border-arc of the front opening. The necessary rotation of ca. 90° of the scaling direction has an additional disadvantage. It nearly disables an additional opening on the rear side, because each mesh of an enlargement there on the left of the dividing curve (21) consequently would be an irregular quadrilateral instead of a trapezoid.
Translational shells and scale-trans shells together are called “TST-shells” hereinafter because, according to the respective point of view, the translational shell, geometrically, is a more symmetric special case of the scale-trans shell, as well as the scale-trans shell, as a more advanced technical development, is a special case of the translational shell.
The known TST-shells are grid shells being glazed with flat panes. The design of their nodes often corresponds to DE 37 15 228 C 2, FIGS. 4, 5. Meanwhile, many variants of it have arisen, enabling subdivisions being more wide-meshed or acute angled. (R. Lehmann: “Knotensteifigkeit von Tragwerken” in: Sophia und Stefan Behling (Ed.): “Glas—Konstruktion und Technologie in der Architektur/Glass—Structure and Technology in Architecture”, München, 1999, p. 74-77, Figs. on p. 75 and 77, each top left). Most of these shells are used for exceptional courtyards and conservatories only. Consequently, they form neither closed spaces nor independent spaces being accessible or usable without additional structures.
TST-shells of mesh elements that transfer loads within the area of the mesh, such as sandwich-panels, are unknown hitherto.